Complex Mapping - Is transforming boundaries enough?

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The discussion revolves around the transformation of regions in the complex plane, specifically from the z-plane to the w-plane, and the implications for boundaries during this process. It highlights the importance of continuous mappings, which preserve the inside-outside relationship, and raises questions about why transformed regions maintain their boundaries. A counterexample is provided, illustrating that not all transformations guarantee that a region will remain inside its transformed boundary, particularly when using certain functions like w = 1/z. The conversation emphasizes the role of analytic transformations as "open maps," yet acknowledges that the behavior depends significantly on the specific contours involved. Ultimately, the relationship between the original and transformed regions is complex and requires careful consideration of the mapping functions.
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Say I will make the transformation from the ##z## plane to the ##w## plane. Moreover, I'll transform a region ##R## with boundary ##C## in the ##z## plane to something in the ##w## plane.

Why is it that if I know the equations for ##C## then I can transform these and immediately know that ##R## will be inside the transformed boundaries? Why isn't it the case that some point in ##R## maps into some point not inside ##C'## (which is my transformed boundary)?

Thanks.
 
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You have to say something about the transformation. Normally, you are using linear, or at least continuous mappings which by definition will preserve this inside-outside relationship.
 
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According to what you say I think continuous mappings would be what I'm dealing with. Strangely I didn't think of continuity, which makes this more intuitive, but not obvious to me.

I see that every point in the neighborhood of ##z_o## in ##R## will also be mapped in ##R'## as a simple closed region. I guess I can visualize these small neighborhoods expanding until they reach the transformed contour...

Is this the argument?

I was told this was because analytic transformations are "open maps" is this what they mean by it?
 
davidbenari said:
Say I will make the transformation from the ##z## plane to the ##w## plane. Moreover, I'll transform a region ##R## with boundary ##C## in the ##z## plane to something in the ##w## plane.

Why is it that if I know the equations for ##C## then I can transform these and immediately know that ##R## will be inside the transformed boundaries? Why isn't it the case that some point in ##R## maps into some point not inside ##C'## (which is my transformed boundary)?

Thanks.
Counterexample: Let C= (z:\lvert z \lvert = 1), then R= (z:\lvert z \lvert < 1). Now transform this using w=\frac{1}{z}. This maps C onto C, but R is mapped to the outside of C.
 
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Svein, but if the function ##w## is analytic in every point in ##R## then what I say will hold right?
 
davidbenari said:
Svein, but if the function ww is analytic in every point in RR then what I say will hold right?
Sorry, I do not know. I checked out Ahlfors, and he carefully does not make that statement. It depends very much on C and the concept of inside.
 
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